How many divisors of 63 are also divisors of 72? (Recall that divisors may be positive or negative.)
First, we list the factors of 63.  They are -63, -21, -9, -7, -3, -1, 1, 3, 7, 9, 21, and 63.  Then, we list the factors of 72.  They are -72, -36, -24, -18, -12, -9, -8, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.  By comparing the two lists, we can see that the divisors that 63 and 72 share are -9, -3, -1, 1, 3, and 9.  (Notice that these six numbers are the divisors of 9; is this a coincidence?) Therefore, 63 and 72 share $\boxed{6}$ divisors.